Mt. Douglas Secondary

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1 Foundations of Math 11 Section 3.4 pplied Problems pplied Problems The Law of Sines and the Law of Cosines are particularly useful for solving applied problems. Please remember when using the Law of Cosines in an SSS situation to find the largest angle first and in SS, after finding the missing side, find the smallest angle first. Example 1 (Surveying) To measure the length of a lake, a baseline is established and measured to be 130 m. ngles and are measured to be 42 and 125 respectively. How long is the lake? Solution: m 42 d Lake C Find C, then use the Law of Sines. C = sin = 13 sin 42 = d 130sin 42 sin13 = 387 metres d = The lake is 387 metres long. Example 2 (Navigation) The course for a boat race starts at point, and heads in a direction S50 W to point, then in a direction S44 E to point C, and finally back north to point. The distance from to C is 10 km. Find the total distance of the boat race. Solution: Since D and C are parallel, C = = D km C = 86 Use Law of Sines. sin86 10 sin86 10 = sin50 a = sin 44 c a = 10sin50 sin86 10sin 44 c = sin86 = 7.68 km = 6.96 km oat race length is = 24.6 km.

2 152 Chapter 3 Non-Right ngle Triangles Foundations of Math 11 Example 3 (Distance) ship is heading due east and passes rock. t this time, the bearing to a lighthouse L is N60 E. fter travelling 5 km, the bearing is N40 E. How far is the ship from the lighthouse? Solution: L = 30, = 130 and L = 20. Use Law of Sines a sin 30 a = sin 20 5 a = 5sin30 sin 20 = 7.3 km The ship is 7.3 km from the lighthouse. 5 km Example 4 (rea) The length of the sides of a triangular parcel of land are approximately 300 m, 400 m and 600 m. pproximate the area of the parcel of land. Solution: Find by Law of Cosines. 300 m h 400 m = (300)(600)cos cos = m C = sin36.34 = h 300 h = 300sin36.34 rea = 1 base height 2 = 1 2 (600)(177.76) = m = m 2 The parcel of land is approximately m 2. Example 5 (Navigation) plane flies 840 km from to at a bearing of N75 E. Then it flies 600 km from to C with a bearing of N30 E. Find the distance from C to. Solution: C D = 15, thus = 135 b km Use Law of Cosines. b 2 = (840)(600)cos km D b = 1334 The distance from C to is 1334 km.

3 Foundations of Math 11 Section 3.4 pplied Problems 153 Example 6 (Surveying) To approximate the length of a lake, a surveyor triangulates the distance to one side to be 950 m and to the other 800 m. If the angle between the two measures is 100, how long is the lake? Solution: Use Law of Cosines. d 2 = (950)(800)cos100 d d = 1344 The lake is 1344 m long. Example 7 (ngle) The distance from home plate to centre field at Yankee Stadium is 400 ft. What is the angle between short stop (half way between 2 nd and 3 rd base) and home plate? S 90 ft 400 ft 90 ft H 90 ft 90 ft x H x 2 = x = 127 feet = = 273 feet Solution: y Law of Cosine. S d ft d 2 = (45)(273)cos135 d = ft y Law of Sine Sin 45 = Sin Sin = = 6.0 The angle between short stop and centre field is 6.0.

4 154 Chapter 3 Non-Right ngle Triangles Foundations of Math Exercise Set 1. Find the length, l, of the brace required to support the lamp. 3 m 2 m plane is sighted by two observers 1 km apart at angles 74 and 78. How high is the plane? km 3. hot air balloon is flying directly between two cities that are 4 km apart. The balloonist finds that the angle of depression to one city is 38 and 33 to the other city. How high above the ground is the balloon? 4. Two planes leave airport at the same time in different directions. One plane lands at airport, 630 km from airport. The other plane lands at airport C some time later. If C = 110 and C = 40, how far did the second plane fly?

5 Foundations of Math 11 Section 3.4 pplied Problems In a solar system, the distance from the Sun (S) to planets and are 85 and 61 million miles respectively. When = 20, how far is it from planet to planet? ' S Three circles with radius = 4 cm, = 3 cm and C = 5 cm are shown. If C = 35, how far is it from the centre of circle to the centre of circle C? C 7. plane flies 420 km from point at a direction of 135 from due east, and then travels 240 km at a direction of 240 from due east. How far is the plane from point? 8. Two planes leave Victoria at 9:00 a.m. One plane travels due east at 500 km/h, while the other plane travels 640 km/h N30 W. How far apart are the two planes at noon?

6 156 Chapter 3 Non-Right ngle Triangles Foundations of Math Two adjacent sides of a parallelogram meet at an angle of 38, and have lengths of 4 cm and 9 cm. What is the length of the larger diagonal of the parallelogram? 10. Three circles of radius 3, 5 and 7 cm are tangent to each other. Find the largest angle formed by joining their centres. 11. The rectangular box has dimensions 4cm 3cm 2cm. Find angle θ formed by a diagonal of the base, and a diagonal of the 2cm 3cm side. 4 cm 3 cm 2 cm 12. n irregular plot of land has dimension as shown. Find. 350 m m 150 m

7 Foundations of Math 11 Section 3.4 pplied Problems fire at C is spotted from two fire lookout stations, and, which are 12 km apart. If station reports C is 50, and station reports C is 32, how far is the fire from station? 14. regular octagon is inscribed in a circle at radius 12 cm. What is the perimeter of the octagon? 15. ship sails from port 50 km on a bearing of 20, then 30 km further on a bearing of 80. How far is the ship from the port? 16. baseball diamond is a square of sides 90 feet, with 60 feet the distance between the pitcher s mound and home plate. When a runner is halfway between second and third base, how far is the runner from the pitcher s mound? field 2nd base pitcher 3rd base 1st base home plate aseball Diamond

8 158 Chapter 3 Non-Right ngle Triangles Foundations of Math The flying distance between Vancouver and Calgary is 675 km. pilot, after flying 240 km from Vancouver, finds she is 6 off course. How far is she from Calgary at this time? 18. On an engine the crankshaft is 8 cm long and the connecting rod, P, is 25 cm long. t the time when PO is 15, how far is the piston P from centre O of the crankshaft? 8 O P P

9 Foundations of Math 11 Section 3.5 Chapter Review Chapter Review Section Find each ratio to four decimal places. a) sin 78 b) cos 41 c) tan 19 d) sin 23.7 e) cos 24.3 f) tan Find the measure of angle α to one decimal place. a) sinα = b) cosα = c) tanα = d) sinα = e) cosα = f) tanα =

10 160 Chapter 3 Non-Right ngle Triangles Foundations of Math Solve the triangles. a) 3 b) x x = 27º 7 α = 9 y x = x y = θ = θ = 4. Find the length of, to the nearest tenth. a) cm b) 10 mm 17 65

11 Foundations of Math 11 Section 3.5 Chapter Review 161 Section Find the angle which gives the same value of sine as the following, 0 θ 180. a) sin 30 b) sin Determine the value of a that will give 0, 1 or 2 triangles. a C a) 0 solutions b) 1 solution c) 2 solutions

12 162 Chapter 3 Non-Right ngle Triangles Foundations of Math Determine if the following leads to 0, 1 or 2 triangles. a) ΔC, = 19, a = 25, C = 30 b) ΔC, = 28, a = 50, b = 20 c) ΔXYZ, X = 58, x = 9.3, z = 6.8 d) ΔXYZ, X = 110, x = 90, z = Solve ΔC using the Law of Sines. a) = 65, = 93, c = 10 b) = 54, b = 9, c = 10

13 Foundations of Math 11 Section 3.5 Chapter Review 163 Section Solve using the Law of Sines or the Law of Cosines. a) = 25, b = 8, c = 14 b) a = 3, b = 4, c = 6 c) C = 60, a = 2, b = 3 d) = 40, = 20, a = 2 e) a = 6, b = 8, = 35 f) a = 2, c = 1, C = 50

14 164 Chapter 3 Non-Right ngle Triangles Foundations of Math Find the area of ΔC. a) a = 2, b = 3, C = 60 b) a = 4, b = 2, c = 5 Section Find the angle from the origin O, between point with coordinate (3, 4) and point with coordinate (4, 3). (angle O) 12. baseball diamond is 90 square feet. If the distance from home plate to straight away centre field is 400 ft, how far is it from first base to centre field?

15 Foundations of Math 11 Section 3.5 Chapter Review softball field is 60 square feet with the pitching rubber 46 feet from home plate. How far is it from the pitching rubber to first base? 14. Three circles of radius 2, 4 and 6 cm are tangent to one another. Find the three angles formed by the lines joining their centres. 15. Find the perimeter of a regular pentagon inscribed by a circle of radius 10 cm. 16. Find the perimeter of a regular pentagon which contains an inscribed circle of radius 10 cm.

16 166 Chapter 3 Non-Right ngle Triangles Foundations of Math 11